Duality-enabled unification of diffusion and wave control via Laplacian constitutive design

Diffusion and wave equations govern two fundamental classes of physical dynamics, yet their controls typically require separate frameworks. Although both reduce to the Laplace equation in the static regime, conventional approaches rely on the quasistatic assumption to control dynamic fields, which often leads to undesired scattering. By establishing dualities among apparently uncorrelated systems, diffusion and wave controls are recast as a unified constitutive design problem governed by the Laplace equation. A potential-eikonal duality is proposed to translate a Laplacian constitutive profile into the precise control of thermal, acoustic, and electromagnetic fields. A force-flow duality, arising from two-dimensional Hodge duality, is revealed to advance multidirectional functional adaptation. This duality-assisted model overcomes the quasistatic and geometric constraints of traditional schemes, enabling freeform cloaking and transparency in either Hermitian or non-Hermitian media. This work provides a general route for multiphysics field control and lays a foundation for scalable metamaterial design.